Given a metric $d$ on a space $X$, what can we say about $d(X\times X)$? What possible range can $d$ have?

More precisely, consider the set $D=\{ A \subset [0,\infty) | A = d(X\times X), \textrm{d is a metric on $X$} \} $ What properties does $D$ have?

For instance, all finite sets containing $0$ are in $D$: If $A=\{0,a_1,a_2...a_n \}$, where $a_i<a_j$ if $i<j$, take $X=\{ 0,1... n \}$ and $d(i,j)= a_{max(i,j)} $ (for $i\neq j$). Any well-ordered countable set is also in $D$, by a similar construction.

1 Answer
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First note that $\varnothing\in D$ by taking $X=\varnothing$.
Next, note that if $A\in D$ is not empty, then $0\in A$.

Now let $A\subseteq [0,\infty)$ such that $0\in A$ and let $d\colon A\times A\to A$ be defined by
$$d(x,y) = \begin{cases} \max(x,y) & x\neq y \\ 0 & x=y \end{cases}.$$
It is not hard to see that $d$ is a metric and the range of $d$ is $A$.